Proofs of trigonometric identities

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Proofs of trigonometric identities are used to show relations between trigonometric functions. This article gives proofs of some trigonometric identities.

Video Proofs of trigonometric identities

Elementary trigonometric identities

Definitions

Referring to the diagram at the right, the six trigonometric functions of ? are:

${\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}={\frac {a}{h}}}$

${\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}={\frac {b}{h}}}$

${\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}={\frac {a}{b}}}$

${\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}={\frac {b}{a}}}$

${\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}={\frac {h}{b}}}$

${\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}={\frac {h}{a}}}$

Ratio identities

The following identities are trivial algebraic consequences of these definitions and the division identity.
They rely on multiplying or dividing the numerator and denominator of fractions by a variable. Ie,

${\displaystyle {\frac {a}{b}}={\frac {\left({\frac {a}{h}}\right)}{\left({\frac {b}{h}}\right)}}}$

${\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}={\frac {\left({\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}\right)}{\left({\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}\right)}}={\frac {\sin \theta }{\cos \theta }}}$

${\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}={\frac {\left({\frac {\mathrm {adjacent} }{\mathrm {adjacent} }}\right)}{\left({\frac {\mathrm {opposite} }{\mathrm {adjacent} }}\right)}}={\frac {1}{\tan \theta }}={\frac {\cos \theta }{\sin \theta }}}$

${\displaystyle \sec \theta ={\frac {1}{\cos \theta }}={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}}$

${\displaystyle \csc \theta ={\frac {1}{\sin \theta }}={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}}$

${\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}={\frac {\left({\frac {\mathrm {opposite} \times \mathrm {hypotenuse} }{\mathrm {opposite} \times \mathrm {adjacent} }}\right)}{\left({\frac {\mathrm {adjacent} \times \mathrm {hypotenuse} }{\mathrm {opposite} \times \mathrm {adjacent} }}\right)}}={\frac {\left({\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}\right)}{\left({\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}\right)}}={\frac {\sec \theta }{\csc \theta }}}$

Or

${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {\left({\frac {1}{\csc \theta }}\right)}{\left({\frac {1}{\sec \theta }}\right)}}={\frac {\left({\frac {\csc \theta \sec \theta }{\csc \theta }}\right)}{\left({\frac {\csc \theta \sec \theta }{\sec \theta }}\right)}}={\frac {\sec \theta }{\csc \theta }}}$

${\displaystyle \cot \theta ={\frac {\csc \theta }{\sec \theta }}}$

Complementary angle identities

Two angles whose sum is ?/2 radians (90 degrees) are complementary. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change ? to ?/2 - ?, obtaining:

${\displaystyle \sin \left(\pi /2-\theta \right)=\cos \theta }$

${\displaystyle \cos \left(\pi /2-\theta \right)=\sin \theta }$

${\displaystyle \tan \left(\pi /2-\theta \right)=\cot \theta }$

${\displaystyle \cot \left(\pi /2-\theta \right)=\tan \theta }$

${\displaystyle \sec \left(\pi /2-\theta \right)=\csc \theta }$

${\displaystyle \csc \left(\pi /2-\theta \right)=\sec \theta }$

Pythagorean identities

Identity 1:

${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}$

The following two results follow from this and the ratio identities. To obtain the first, divide both sides of ${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}$ by ${\displaystyle \cos ^{2}(x)}$; for the second, divide by ${\displaystyle \sin ^{2}(x)}$.

${\displaystyle \tan ^{2}(x)+1\ =\sec ^{2}(x)}$

${\displaystyle 1\ +\cot ^{2}(x)=\csc ^{2}(x)}$

Similarly

${\displaystyle 1\ +\cot ^{2}(x)=\csc ^{2}(x)}$

${\displaystyle \csc ^{2}(x)-\cot ^{2}(x)=1}$

Identity 2:

The following accounts for all three reciprocal functions.

${\displaystyle \csc ^{2}(x)+\sec ^{2}(x)-\cot ^{2}(x)=2\ +\tan ^{2}(x)}$

Proof 2:

Refer to the triangle diagram above. Note that ${\displaystyle a^{2}+b^{2}=h^{2}}$ by Pythagorean theorem.

${\displaystyle \csc ^{2}(x)+\sec ^{2}(x)={\frac {h^{2}}{a^{2}}}+{\frac {h^{2}}{b^{2}}}={\frac {a^{2}+b^{2}}{a^{2}}}+{\frac {a^{2}+b^{2}}{b^{2}}}=2\ +{\frac {b^{2}}{a^{2}}}+{\frac {a^{2}}{b^{2}}}}$

Substituting with appropriate functions -

${\displaystyle 2\ +{\frac {b^{2}}{a^{2}}}+{\frac {a^{2}}{b^{2}}}=2\ +\tan ^{2}(x)+\cot ^{2}(x)}$

Rearranging gives:

${\displaystyle \csc ^{2}(x)+\sec ^{2}(x)-\cot ^{2}(x)=2\ +\tan ^{2}(x)}$

Angle sum identities

Sine

Draw a horizontal line (the x-axis); mark an origin O. Draw a line from O at an angle ${\displaystyle \alpha }$ above the horizontal line and a second line at an angle ${\displaystyle \beta }$ above that; the angle between the second line and the x-axis is ${\displaystyle \alpha +\beta }$.

Place P on the line defined by ${\displaystyle \alpha +\beta }$ at a unit distance from the origin.

Let PQ be a line perpendicular to line defined by angle ${\displaystyle \alpha }$, drawn from point Q on this line to point P. ${\displaystyle \therefore }$ OQP is a right angle.

Let QA be a perpendicular from point A on the x-axis to Q and PB be a perpendicular from point B on the x-axis to P. ${\displaystyle \therefore }$ OAQ and OBP are right angles.

Draw R on PB so that QR is parallel to the x-axis.

Now angle ${\displaystyle RPQ=\alpha }$ (because ${\displaystyle OQA={\frac {\pi }{2}}-\alpha }$, making ${\displaystyle RQO=\alpha ,RQP={\frac {\pi }{2}}-\alpha }$, and finally ${\displaystyle RPQ=\alpha }$)

${\displaystyle RPQ={\tfrac {\pi }{2}}-RQP={\tfrac {\pi }{2}}-({\tfrac {\pi }{2}}-RQO)=RQO=\alpha }$

${\displaystyle OP=1}$

${\displaystyle PQ=\sin \beta }$

${\displaystyle OQ=\cos \beta }$

${\displaystyle {\frac {AQ}{OQ}}=\sin \alpha }$, so ${\displaystyle AQ=\sin \alpha \cos \beta }$

${\displaystyle {\frac {PR}{PQ}}=\cos \alpha }$, so ${\displaystyle PR=\cos \alpha \sin \beta }$

${\displaystyle \sin(\alpha +\beta )=PB=RB+PR=AQ+PR=\sin \alpha \cos \beta +\cos \alpha \sin \beta }$

By substituting ${\displaystyle -\beta }$ for ${\displaystyle \beta }$ and using Symmetry, we also get:

${\displaystyle \sin(\alpha -\beta )=\sin \alpha \cos(-\beta )+\cos \alpha \sin(-\beta )}$

${\displaystyle \sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta }$

Another rigorous proof, and much easier, can be given by using Euler's formula, known from complex analysis. Euler's formula is:

${\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi }$

It follows that for angles ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ we have:

${\displaystyle e^{i(\alpha +\beta )}=\cos(\alpha +\beta )+i\sin(\alpha +\beta )}$

Also using the following properties of exponential functions:

${\displaystyle e^{i(\alpha +\beta )}=e^{i\alpha }e^{i\beta }=(\cos \alpha +i\sin \alpha )(\cos \beta +i\sin \beta )}$

Evaluating the product:

${\displaystyle e^{i(\alpha +\beta )}=(\cos \alpha \cos \beta -\sin \alpha \sin \beta )+i(\sin \alpha \cos \beta +\sin \beta \cos \alpha )}$

Equating real and imaginary parts:

${\displaystyle \cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta }$

${\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\sin \beta \cos \alpha }$

Cosine

Using the figure above,

${\displaystyle OP=1}$

${\displaystyle PQ=\sin \beta }$

${\displaystyle OQ=\cos \beta }$

${\displaystyle {\frac {OA}{OQ}}=\cos \alpha }$, so ${\displaystyle OA=\cos \alpha \cos \beta }$

${\displaystyle {\frac {RQ}{PQ}}=\sin \alpha }$, so ${\displaystyle RQ=\sin \alpha \sin \beta }$

${\displaystyle \cos(\alpha +\beta )=OB=OA-BA=OA-RQ=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta }$

By substituting ${\displaystyle -\beta }$ for ${\displaystyle \beta }$ and using Symmetry, we also get:

${\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos(-\beta )-\sin \alpha \sin(-\beta ),}$

${\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta }$

Also, using the complementary angle formulae,

${\displaystyle {\begin{aligned}\cos(\alpha +\beta )&=\sin \left(\pi /2-(\alpha +\beta )\right)\\&=\sin \left((\pi /2-\alpha )-\beta \right)\\&=\sin \left(\pi /2-\alpha \right)\cos \beta -\cos \left(\pi /2-\alpha \right)\sin \beta \\&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\end{aligned}}}$

Tangent and cotangent

From the sine and cosine formulae, we get

${\displaystyle \tan(\alpha +\beta )={\frac {\sin(\alpha +\beta )}{\cos(\alpha +\beta )}}={\frac {\sin \alpha \cos \beta +\cos \alpha \sin \beta }{\cos \alpha \cos \beta -\sin \alpha \sin \beta }}}$

Dividing both numerator and denominator by ${\displaystyle \cos \alpha \cos \beta }$, we get

${\displaystyle \tan(\alpha +\beta )={\frac {\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }}}$

Subtracting ${\displaystyle \beta }$ from ${\displaystyle \alpha }$, using ${\displaystyle \tan(-\beta )=-\tan \beta }$,

${\displaystyle \tan(\alpha -\beta )={\frac {\tan \alpha +\tan(-\beta )}{1-\tan \alpha \tan(-\beta )}}={\frac {\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }}}$

Similarly from the sine and cosine formulae, we get

${\displaystyle \cot(\alpha +\beta )={\frac {\cos(\alpha +\beta )}{\sin(\alpha +\beta )}}={\frac {\cos \alpha \cos \beta -\sin \alpha \sin \beta }{\sin \alpha \cos \beta +\cos \alpha \sin \beta }}}$

Then by dividing both numerator and denominator by ${\displaystyle \sin \alpha \sin \beta }$, we get

${\displaystyle \cot(\alpha +\beta )={\frac {\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }}}$

Or, using ${\displaystyle \cot \theta ={\frac {1}{\tan \theta }}}$,

${\displaystyle \cot(\alpha +\beta )={\frac {1-\tan \alpha \tan \beta }{\tan \alpha +\tan \beta }}={\frac {{\frac {1}{\tan \alpha \tan \beta }}-1}{{\frac {1}{\tan \alpha }}+{\frac {1}{\tan \beta }}}}={\frac {\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }}}$

Using ${\displaystyle \cot(-\beta )=-\cot \beta }$,

${\displaystyle \cot(\alpha -\beta )={\frac {\cot \alpha \cot(-\beta )-1}{\cot \alpha +\cot(-\beta )}}={\frac {\cot \alpha \cot \beta +1}{\cot \beta -\cot \alpha }}}$

Double-angle identities

From the angle sum identities, we get

${\displaystyle \sin(2\theta )=2\sin \theta \cos \theta }$

and

${\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta }$

The Pythagorean identities give the two alternative forms for the latter of these:

${\displaystyle \cos(2\theta )=2\cos ^{2}\theta -1}$

${\displaystyle \cos(2\theta )=1-2\sin ^{2}\theta }$

The angle sum identities also give

${\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}={\frac {2}{\cot \theta -\tan \theta }}}$

${\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}={\frac {\cot \theta -\tan \theta }{2}}}$

It can also be proved using Euler's formula

${\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi }$

Squaring both sides yields

${\displaystyle e^{i2\varphi }=(\cos \varphi +i\sin \varphi )^{2}}$

But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields

${\displaystyle e^{i2\varphi }=\cos 2\varphi +i\sin 2\varphi }$

It follows that

${\displaystyle (\cos \varphi +i\sin \varphi )^{2}=\cos 2\varphi +i\sin 2\varphi }$.

Expanding the square and simplifying on the left hand side of the equation gives

${\displaystyle i(2\sin \varphi \cos \varphi )+\cos ^{2}\varphi -\sin ^{2}\varphi \ =\cos 2\varphi +i\sin 2\varphi }$.

Because the imaginary and real parts have to be the same, we are left with the original identities

${\displaystyle \cos ^{2}\varphi -\sin ^{2}\varphi \ =\cos 2\varphi }$,

and also

${\displaystyle 2\sin \varphi \cos \varphi =\sin 2\varphi }$.

Half-angle identities

The two identities giving the alternative forms for cos 2? lead to the following equations:

${\displaystyle \cos {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}},}$

${\displaystyle \sin {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}.}$

The sign of the square root needs to be chosen properly--note that if ? is added to ?, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of ?.

For the tan function, the equation is:

${\displaystyle \tan {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}.}$

Then multiplying the numerator and denominator inside the square root by (1 + cos ?) and using Pythagorean identities leads to:

${\displaystyle \tan {\frac {\theta }{2}}={\frac {\sin \theta }{1+\cos \theta }}.}$

Also, if the numerator and denominator are both multiplied by (1 - cos ?), the result is:

${\displaystyle \tan {\frac {\theta }{2}}={\frac {1-\cos \theta }{\sin \theta }}.}$

This also gives:

${\displaystyle \tan {\frac {\theta }{2}}=\csc \theta -\cot \theta .}$

Similar manipulations for the cot function give:

${\displaystyle \cot {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}={\frac {1+\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1-\cos \theta }}=\csc \theta +\cot \theta .}$

Miscellaneous -- the triple tangent identity

If ${\displaystyle \psi +\theta +\phi =\pi =}$ half circle (for example, ${\displaystyle \psi }$, ${\displaystyle \theta }$ and ${\displaystyle \phi }$ are the angles of a triangle),

${\displaystyle \tan(\psi )+\tan(\theta )+\tan(\phi )=\tan(\psi )\tan(\theta )\tan(\phi ).}$

Proof:

${\displaystyle {\begin{aligned}\psi &=\pi -\theta -\phi \\\tan(\psi )&=\tan(\pi -\theta -\phi )\\&=-\tan(\theta +\phi )\\&={\frac {-\tan \theta -\tan \phi }{1-\tan \theta \tan \phi }}\\&={\frac {\tan \theta +\tan \phi }{\tan \theta \tan \phi -1}}\\(\tan \theta \tan \phi -1)\tan \psi &=\tan \theta +\tan \phi \\\tan \psi \tan \theta \tan \phi -\tan \psi &=\tan \theta +\tan \phi \\\tan \psi \tan \theta \tan \phi &=\tan \psi +\tan \theta +\tan \phi \\\end{aligned}}}$

Miscellaneous -- the triple cotangent identity

If ${\displaystyle \psi +\theta +\phi ={\tfrac {\pi }{2}}=}$ quarter circle,

${\displaystyle \cot(\psi )+\cot(\theta )+\cot(\phi )=\cot(\psi )\cot(\theta )\cot(\phi )}$.

Proof:

Replace each of ${\displaystyle \psi }$, ${\displaystyle \theta }$, and ${\displaystyle \phi }$ with their complementary angles, so cotangents turn into tangents and vice versa.

Given

${\displaystyle \psi +\theta +\phi ={\tfrac {\pi }{2}}}$

${\displaystyle \therefore ({\tfrac {\pi }{2}}-\psi )+({\tfrac {\pi }{2}}-\theta )+({\tfrac {\pi }{2}}-\phi )={\tfrac {3\pi }{2}}-(\psi +\theta +\phi )={\tfrac {3\pi }{2}}-{\tfrac {\pi }{2}}=\pi }$

so the result follows from the triple tangent identity.

Prosthaphaeresis identities

• ${\displaystyle \sin \theta \pm \sin \phi =2\sin \left({\frac {\theta \pm \phi }{2}}\right)\cos \left({\frac {\theta \mp \phi }{2}}\right)}$
• ${\displaystyle \cos \theta +\cos \phi =2\cos \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)}$
• ${\displaystyle \cos \theta -\cos \phi =-2\sin \left({\frac {\theta +\phi }{2}}\right)\sin \left({\frac {\theta -\phi }{2}}\right)}$

Proof of sine identities

First, start with the sum-angle identities:

${\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta }$

${\displaystyle \sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta }$

By adding these together,

${\displaystyle \sin(\alpha +\beta )+\sin(\alpha -\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta +\sin \alpha \cos \beta -\cos \alpha \sin \beta =2\sin \alpha \cos \beta }$

Similarly, by subtracting the two sum-angle identities,

${\displaystyle \sin(\alpha +\beta )-\sin(\alpha -\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta -\sin \alpha \cos \beta +\cos \alpha \sin \beta =2\cos \alpha \sin \beta }$

Let ${\displaystyle \alpha +\beta =\theta }$ and ${\displaystyle \alpha -\beta =\phi }$,

${\displaystyle \therefore \alpha ={\frac {\theta +\phi }{2}}}$ and ${\displaystyle \beta ={\frac {\theta -\phi }{2}}}$

Substitute ${\displaystyle \theta }$ and ${\displaystyle \phi }$

${\displaystyle \sin \theta +\sin \phi =2\sin \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)}$

${\displaystyle \sin \theta -\sin \phi =2\cos \left({\frac {\theta +\phi }{2}}\right)\sin \left({\frac {\theta -\phi }{2}}\right)=2\sin \left({\frac {\theta -\phi }{2}}\right)\cos \left({\frac {\theta +\phi }{2}}\right)}$

Therefore,

${\displaystyle \sin \theta \pm \sin \phi =2\sin \left({\frac {\theta \pm \phi }{2}}\right)\cos \left({\frac {\theta \mp \phi }{2}}\right)}$

Proof of cosine identities

Similarly for cosine, start with the sum-angle identities:

${\displaystyle \cos(\alpha +\beta )=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta }$

${\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta }$

Again, by adding and subtracting

${\displaystyle \cos(\alpha +\beta )+\cos(\alpha -\beta )=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta +\cos \alpha \cos \beta +\sin \alpha \sin \beta =2\cos \alpha \cos \beta }$

${\displaystyle \cos(\alpha +\beta )-\cos(\alpha -\beta )=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta -\cos \alpha \cos \beta -\sin \alpha \sin \beta =-2\sin \alpha \sin \beta }$

Substitute ${\displaystyle \theta }$ and ${\displaystyle \phi }$ as before,

${\displaystyle \cos \theta +\cos \phi =2\cos \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)}$

${\displaystyle \cos \theta -\cos \phi =-2\sin \left({\frac {\theta +\phi }{2}}\right)\sin \left({\frac {\theta -\phi }{2}}\right)}$

Inequalities

The figure at the right shows a sector of a circle with radius 1. The sector is ?/(2?) of the whole circle, so its area is ?/2. We assume here that ? < ?/2.

${\displaystyle OA=OD=1}$

${\displaystyle AB=\sin \theta }$

${\displaystyle CD=\tan \theta }$

The area of triangle OAD is AB/2, or sin(?)/2. The area of triangle OCD is CD/2, or tan(?)/2.

Since triangle OAD lies completely inside the sector, which in turn lies completely inside triangle OCD, we have

${\displaystyle \sin \theta <\theta <\tan \theta .}$

This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. For the sine function, we can handle other values. If ? > ?/2, then ? > 1. But sin ? <= 1 (because of the Pythagorean identity), so sin ? < ?. So we have

${\displaystyle {\frac {\sin \theta }{\theta }}<1\ \ \ \mathrm {if} \ \ \ 0<\theta .}$

For negative values of ? we have, by symmetry of the sine function

${\displaystyle {\frac {\sin \theta }{\theta }}={\frac {\sin(-\theta )}{-\theta }}<1.}$

Hence

${\displaystyle {\frac {\sin \theta }{\theta }}<1\quad {\text{if }}\quad \theta \neq 0,}$

and

${\displaystyle {\frac {\tan \theta }{\theta }}>1\quad {\text{if }}\quad 0<\theta <{\frac {\pi }{2}}.}$

Maps Proofs of trigonometric identities

Identities involving calculus

Preliminaries

${\displaystyle \lim _{\theta \to 0}{\sin \theta }=0}$

${\displaystyle \lim _{\theta \to 0}{\cos \theta }=1}$

Sine and angle ratio identity

${\displaystyle \lim _{\theta \to 0}{\frac {\sin \theta }{\theta }}=1}$

Proof: From the previous inequalities, we have, for small angles

${\displaystyle \sin \theta <\theta <\tan \theta }$,

Therefore,

${\displaystyle {\frac {\sin \theta }{\theta }}<1<{\frac {\tan \theta }{\theta }}}$,

Consider the right-hand inequality. Since

${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}}$

${\displaystyle \therefore 1<{\frac {\sin \theta }{\theta \cos \theta }}}$

Multiply through by ${\displaystyle \cos \theta }$

${\displaystyle \cos \theta <{\frac {\sin \theta }{\theta }}}$

Combining with the left-hand inequality:

${\displaystyle \cos \theta <{\frac {\sin \theta }{\theta }}<1}$

Taking ${\displaystyle \cos \theta }$ to the limit as ${\displaystyle \theta \to 0}$

${\displaystyle \lim _{\theta \to 0}{\cos \theta }=1}$

Therefore,

${\displaystyle \lim _{\theta \to 0}{\frac {\sin \theta }{\theta }}=1}$

Cosine and angle ratio identity

${\displaystyle \lim _{\theta \to 0}{\frac {1-\cos \theta }{\theta }}=0}$

Proof:

${\displaystyle {\begin{aligned}{\frac {1-\cos \theta }{\theta }}&={\frac {1-\cos ^{2}\theta }{\theta (1+\cos \theta )}}\\&={\frac {\sin ^{2}\theta }{\theta (1+\cos \theta )}}\\&=\left({\frac {\sin \theta }{\theta }}\right)\times \sin \theta \times \left({\frac {1}{1+\cos \theta }}\right)\\\end{aligned}}}$

The limits of those three quantities are 1, 0, and 1/2, so the resultant limit is zero.

Cosine and square of angle ratio identity

${\displaystyle \lim _{\theta \to 0}{\frac {1-\cos \theta }{\theta ^{2}}}={\frac {1}{2}}}$

Proof:

As in the preceding proof,

${\displaystyle {\frac {1-\cos \theta }{\theta ^{2}}}={\frac {\sin \theta }{\theta }}\times {\frac {\sin \theta }{\theta }}\times {\frac {1}{1+\cos \theta }}.}$

The limits of those three quantities are 1, 1, and 1/2, so the resultant limit is 1/2.

Proof of compositions of trig and inverse trig functions

All these functions follow from the Pythagorean trigonometric identity. We can prove for instance the function

${\displaystyle \sin[\arctan(x)]={\frac {x}{\sqrt {1+x^{2}}}}}$

Proof:

We start from

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$

Then we divide this equation by ${\displaystyle \cos ^{2}\theta }$

${\displaystyle \cos ^{2}\theta ={\frac {1}{\tan ^{2}\theta +1}}}$

Then use the substitution ${\displaystyle \theta =\arctan(x)}$, also use the Pythagorean trigonometric identity:

${\displaystyle 1-\sin ^{2}[\arctan(x)]={\frac {1}{\tan ^{2}[\arctan(x)]+1}}}$

Then we use the identity ${\displaystyle \tan[\arctan(x)]\equiv x}$

${\displaystyle \sin[\arctan(x)]={\frac {x}{\sqrt {x^{2}+1}}}}$

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See also

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Notes

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References

Source of article : Wikipedia